Problem Statement

We have a checkerboard with H rows and W columns, where each square has a 0 or 1 written on it.
The current state of checkerboard is represented by $H strings S_1,S_2,\cdots,S_H$: the j-th character of S_i represents the digit in the square at the i-th row from the top and j-th column from the left.
Snuke will repeat the following operation.

• Choose one square uniformly at random.
  Then, flip the value written in that square. (In other words, change 0 to 1 and vice versa.)

By the way, he loves an integer sequence $A=(A_1,A_2,\cdots,A_H)$, so he will terminate the process at the moment when the following condition is satisfied.

• For every $i (1 \leq i \leq H)$, the i-th row from the top contains exactly A_i 1s.

Particularly, he may do zero operations.
Find the expected value of the number of operations Snuke does, modulo 998244353.
Definition of the expected value modulo 998244353

It can be proved that the sought expected value is always a rational number. Additionally, under the Constraints of this problem, when that value is represented as an irreducible fraction $\frac{P}{Q}$, it can also be proved that $Q \not$ $\equiv 0 \pmod{998244353}. Thus, there uniquely exists an integer R$ such that $R \times Q \equiv P \pmod{998244353}, 0 \leq R < 998244353.$ Find this R.

Input

Input is given from Standard Input in the following format:
H W
S_1
S_2
$\vdots$
S_H
$A_1 A_2 \cdots A_H$

Output

Print the answer.

Constraints

• $1 \leq H,W \leq 50$
• S_i is a string of length W consisting of 0, 1.
• $0 \leq A_i \leq W$

Sample Input 1

1 2
01
0

Sample Output 1

3

The process goes as follows.

• With probability 1/2, a square with 1 is flipped. The 1-st row now contains zero 1s, terminating the process.
• With probability 1/2, a square with 0 is flipped. The 1-st row now contains two 1s, continuing the process.
• One of the squares is flipped. Regardless of which square is flipped, the 1-st row now contains one 1, continuing the process.
• With probability 1/2, a square with 1 is flipped. The 1-st row now contains zero 1s, terminating the process.
• With probability 1/2, a square with 0 is flipped. The 1-st row now contains two 1s, continuing the process.
• $\vdots$

The expected value of the number of operations is 3.

Sample Input 2

3 3
000
100
110
0 1 2

Sample Output 2

0

Sample Input 3

2 2
00
01
1 0

Sample Output 3

332748127

Sample Input 4

5 4
1101
0000
0010
0100
1111
1 3 3 2 1

Sample Output 4

647836743

见题面 Input 部分

见题面 Output 部分